An Essay on the Origins and Development of Nonlinear Dynamics, Chaos and Complex Systems Miguel A. F. Sanjuán* Department of Physics Universidad Rey Juan Carlos 28933 Móstoles, Madrid, Spain Email: [email protected] ABSTRACT We can find dynamics in every field of science, including economics, chemical reactions, physiology or neurodynamics, showing its intrinsic interdisciplinarity. The interactions between the parts of a system and their feedback mechanisms constitute a source of nonlinearity and complexity, which added to the sensitive dependence on initial conditions, hallmark of chaotic behavior, lead to a crucial change of perspective with serious consequences in the understanding of science. Relevant problems such as the three-body problem in celestial mechanics, turbulence in fluid dynamics, irreversibility in statistical physics; or the logistic equation in population dynamics, have been at the origins of this fascinating field of nonlinear dynamics, chaos, and complex systems. A knowledge of its origins, as well as the many schools of mathematics and physics that have contributed to its development, allow us to better understand the discipline and the breadth of its many applications to science. 1. Introduction Nonlinear dynamics is the discipline that aims to study nonlinear dynamical systems, which are those systems defined by one or more variables evolving with time where the response is not proportional to the stimulus. Chaos is one of the three kinds of motion, in addition to the periodic and quasi- periodic motions. Naturally, there are as many dynamical systems as there are variables that have a temporal evolution, which gives us an idea of the interdisciplinary nature and scope of nonlinear dynamics [1-4]. Many of the ideas and concepts of complexity, such as a science of complex systems, pose a real challenge for the integration of various disciplines, among which we must point out nonlinear dynamics and chaos theory, statistical physics, stochastic processes, information theory, network theory, engineering science, life sciences, and computer sciences. This listing is naturally not complete, but it gives an idea of the challenge behind the idea of complexity. This intended goal means more than the idea of crossing disciplinary boundaries, but rather to integrate disciplines in a common background. Much has been discussed in recent years about a fruitful dialogue between different scientific disciplines, not only to solve old problems, but also as a source of inspiration for new problems. For the study of complexity this is one of the fundamental elements, since its object of study covers problems related to both the so-called hard and soft sciences. Complex systems exist in biology, chemistry, physics, sociology, economics, etc. In any case, the true dialogue between disciplines so necessary for the advancement of knowledge of complex systems in particular, and of science in general, is still lacking. Different paths that have led to the understanding of chaos as we understand it today. Among them, I like to point out: (1) The logistic map and population dynamics (2) Nonlinear oscillators (3) The three-body problem in celestial mechanics (4) Turbulence in fluid dynamics and (5) Irreversibility in statistical mechanics. All of them will be discussed throughout this article. 2. Nonlinear Dynamics and Deterministic Chaos As previously discussed, dynamics is the science that studies the variation in time of different variables, that is, its motion. Basically, there are three types of motion: stationary and equilibrium; periodic and quasi-periodic; and finally chaotic motion. Considering the notion of motion in a broad sense, it is easy to understand that we can find dynamical systems in any scientific MIGUEL A. F. SANJUÁN discipline. That is why it is customary to say that one of the characteristics of nonlinear dynamics is its interdisciplinarity, since with its methods we can approach the study of many different phenomena that evolve over time. We use the term "nonlinear" to logically contrast it with the term "linear", since the linear approach is the one traditionally used in science due to its mathematical simplicity. The linear approach implies the assumption of properties such as: (1) Proportionality: small causes cause small effects (2) Additivity: the whole is equal to the sum of its parts (3) Replication: the same action under the same conditions produces the same result and (4) clear relationships between cause and effect: it is enough to know a little about the behavior of a system to fully know it. However, when nature's relationships are not linear, it leads us to very different situations. A proportional relationship between two variables x and y, where y = kx, indicates a linear relationship. Therefore, any relationship between two variables that does not respond to a proportional relationship like the previous one will be nonlinear. It is easy to figure out that most dynamical systems are nonlinear. When there are relationships of nonlinearity, there can be chaotic behavior that has the following properties: (1) There is no proportionality: small causes can cause large effects (2) Emergence: additivity does not exist, so the whole is greater than the sum of its parts (3) Sensitive dependence on initial conditions: which can make that the same experiment can never be reproduced exactly; and finally (4) Nonlinearity that can generate instabilities, discontinuities and unpredictability, which requires flexibility, adaptability, dynamic change, innovation and reaction capacity. Possibly one of the deepest ideas about the nature of what is known as chaotic behavior is the idea of sensitive dependence on the initial conditions. That is, trajectories of a chaotic system move away from each other as time progresses when they start from very close initial points. This fact has very drastic consequences on the predictability of a system. From this viewpoint, it is somehow surprising to read the following sentence from the Chapter XIV of The Origin of Species (1859) [5] by Charles Darwin: "More individuals are born than can possibly survive. A grain in the balance will determine which individuals shall live and which will shall die, which variety or species shall increase in number, and which shall decrease, or finally become extinct", 3 that in a certain sense shows already the true notion of sensitive dependence on initial conditions. In this regard, it is also interesting to bring up a famous rhyme traditionally associated with Benjamin Franklin, although antecedents of the same idea date back to the 15th century, and which is known as "For Want of a Nail " as shown in Fig. 1. Fig. 1: The rhyme "For Want of a Nail..." offers an intuitive and poetic image of the idea of sensitive dependence on initial conditions, which is the hallmark of chaos. We can define chaotic behavior or chaos as a type of motion that is derived from deterministic temporal dynamics of simple systems that can in fact be described in terms of few variables and whose fundamental characteristics are: (1) Being irregular in time, and given its nonlinear character, of course, cannot be the superposition of periodic motions, being in fact of aperiodic nature and bounded (2) Be unpredictable in the long term and very sensitive on the initial conditions and (3) Be complex, but ordered in the phase space, presenting a geometry of a fractal nature. If we compare the chaotic motion with the regular motion, we can say that the latter is repetitive, periodic, predictable and with a simple geometry, while the former is irregular, unpredictable and with a complicated geometry. There are different types of chaotic motions. It is fundamentally called permanent chaos when once a dynamical system finds this state it remains in it forever. On the other hand, it is called transient chaos when this chaotic behavior occurs only in a certain period of time and the system subsequently MIGUEL A. F. SANJUÁN behaves differently. Furthermore, dynamical systems generally distinguish between dissipative and conservative based on whether or not they conserve energy. Well, for dissipative systems permanent chaos occurs in what is called a chaotic attractor in the phase space. However, in the case of transient chaos, chaotic transients occur in a fractal set. In the conservative case, on the one hand, permanent chaos occurs in bounded regions of the phase space and transient chaos is associated, for example, with the phenomenon of chaotic scattering that occurs in numerous physical phenomena, giving rise to very complex fractal structures. These concepts will be explained in more detail throughout the article. Dynamical systems are usually classified as discrete and continuous depending on whether time is measured discretely or continuously. A paradigm for discrete dynamical systems is the logistic map, defined as xn+1 = rxn (1− xn ) , which is an iterative equation where the index n indicates an iteration that is linked to the discrete way of measuring time. Figure 2 shows a Feigenbaum bifurcation diagram corresponding to the logistic map, where the final state of the system is displayed as a function of the variation of parameter r. Fig. 2: Feigenbaum bifurcation diagram corresponding to the logistic map. It indicates how the final state of the system varies depending on the value of parameter r. A paradigm for continuous systems is the simple pendulum (Fig. 3). It consists of a body of mass m that hangs on a cord that is in principle inextensible and of negligible mass, and whose suspension point moves periodically. 5 Fig. 3. Oscillatory motion of a pendulum In this system, time is measured continuously, and therefore it can be modeled, once normalized, using a differential equation such as !x!+ µx! +sin x = F cosωt . This equation contains in addition to the inertia term (the second derivative of the position), the friction of intensity µ which is proportional to the velocity, the nonlinear sinusoidal term and an external periodic forcing of amplitude F and frequency w.